1. The measure of an interior angle of a regular polygon is 20 more than thrice the measure of its adjacent exterior angle. Find the number of sides of the polygon and its total number of diagonals.

Question

1.  The measure of an interior angle of a regular polygon is 20 more than thrice the measure of its adjacent exterior angle.  Find the number of sides of the polygon and its total number of diagonals.

2.  Find the sum and difference between the sum of the measures of the interior angles of a convex 23-gon and 14-gon.

Reiny · Answer

1. let the number of sides be n
each interior angle = 180(n-2)/n
exterior angle = 360/n

180(n-2)/n = 3(360/n) + 20
times n
180(n-2) = 1080 + 20n
180n - 360 = 1080 + 20n
160n = 1440
n = 9

We have a 9-gon, which is called a enneagon or nonagon

check:
interior angle = 180(7)/9 = 140
exterior angle = 360/9 = 40
is 3 times 40 + 20 = 140 ? , YES

2.
sum interior angles of 23-gon= 180(21) = 3780
sum of interior angles of 14-gon = ....
then take their sum and difference

Anonymous · Answer

The measure of each interior angle of a regular polygon is 20 more than three times the measure of each exterior angle. How many sides does the polygon have?

Asna · Answer

If the outer angle is 20 more than thrice the inner angle what is the measure of the inner angle